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Adaptive density estimation under weak dependence

Published online by Cambridge University Press:  10 May 2010

Irène Gannaz  and
Olivier Wintenberger
Irène Gannaz
Affiliation:
Laboratoire Jean Kuntzmann, INP Grenoble, 38041 Grenoble Cedex 9, France
Olivier Wintenberger
Affiliation:
SAMOS-MATISSE (Statistique Appliquée et Modélisation Stochastique), Centre d'Économie de la Sorbonne Université Paris 1 – Panthéon-Sorbonne, CNRS 90, Rue de Tolbiac, 75634 Paris Cedex 13, France
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Abstract

Assume that (Xt)t∈Z is a real valued time series admitting a common marginal density f with respect to Lebesgue's measure. [Donoho et al. Ann. Stat.24 (1996) 508–539] propose near-minimax estimators $\widehat f_n$ based on thresholding wavelets to estimate f on a compact set in an independent and identically distributed setting. The aim of the present work is to extend these results to general weak dependent contexts. Weak dependence assumptions are expressed as decreasing bounds of covariance terms and are detailed for different examples. The threshold levels in estimators $\widehat f_n$ depend on weak dependence properties of the sequence (Xt)t∈Z through the constant. If these properties are unknown, we propose cross-validation procedures to get new estimators. These procedures are illustrated via simulations of dynamical systems and non causal infinite moving averages. We also discuss the efficiency of our estimators with respect to the decrease of covariances bounds.

Keywords

Adaptive estimationcross validationhard thresholdingnear minimax resultsnonparametric density estimationsoft thresholdingwaveletsweak dependence
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 14 , 2010 , pp. 151 - 172
Copyright
© EDP Sciences, SMAI, 2010

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